Range minimum queries in minimal space

Luís M. S. Russo

Abstract: We consider the problem of computing a sequence of range minimum queries. We assume a sequence of commands that contains values and queries. Our goal is to quickly determine the minimum value that exists between the current position and a previous position i. Range minimum queries are used as a sub-routine of several algorithms, namely related to string processing. We propose a data structure that can process these command sequences. We obtain efficient results for several variations of the problem, in particular we obtain O(1)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math> time per command for the offline version and O(α(n))<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">α</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">)</mo></math> amortized time for the online version, where α(n)<math><mi is="true">α</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> is the inverse Ackermann function and n the number of values in the sequence. This data structure also has very small space requirements, namely O(ℓ)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">ℓ</mi><mo stretchy="false" is="true">)</mo></math> where ℓ is the maximum number of active i positions. We implemented our data structure and show that it is competitive against existing alternatives. We obtain comparable processing time, in the nanosecond range, and much smaller space requirements.